3D ROTATION ESTIMATION USING DISCRETE SPHERICAL HARMONIC OSCILLATOR TRANSFORMS Soo-Chang Pei1 and Chun-Lin Liu2 National Taiwan University1; California Institute of Technology2 Department of Electrical Engineering1,2 No. 1, Sec. 4 Roosevolt Rd. Taipei, Taiwan1; 1200 E. California Blvd.,Pasadena, CA 91125, USA2 E-mail: [email protected] and [email protected] ABSTRACT existing angle estimation algorithm and the overall system diagram is summarized in Fig. 2. Section 6 experiments on the estimation This paper presents an approach to 3D rotation estimation using accuracy and the noise robustness of the proposed method, compared discrete spherical harmonic oscillator transforms (discrete SHOTs). to some existing approaches. Section 7 concludes this paper. Discrete SHOTs not only have simple and fast implementation meth- ods but also are compatible with the existing angle estimation algo- rithms related to spherical harmonics. Discrete SHOTs of the ro- 2. RELATION TO PRIOR WORK tated signal follow the same formulation to the Wigner-D matrix as spherical harmonics transforms. Thus, the spherical harmonics re- Our work considers the angle estimation problem when the input ob- lated algorithms could be utilized to discrete SHOTs without modi- jects are sampled uniformly on the 3D Cartesian coordinates. That 3 fication. Furthermore, compared to some existing methods, our ap- is, the input samples are defined on the R grid, also called the vol- proach with discrete SHOTs exhibits higher accuracy, higher preci- ume data, used widely in medical imaging. Most angle estimation sion and improved robustness to noise if the input signal is sampled algorithms with the spherical harmonics work on the surface input uniformly on Cartesian grids. The phenomenon results from no in- data [1,6,7], where the data is parametrized in terms of the polar an- terpolations in discrete SHOTs. gle θ and the azimuthal angle ϕ. The surface information describes the general outline of the object in 2D but the full energy distribution Index Terms — Discrete spherical harmonic oscillator trans- of the object in 3D is not included. forms, spherical harmonic transforms, rotation estimation, Euler The angle estimation algorithm proposed by Burel and H´enocq angles, volume data. [8] dealt with the angle estimation formulae for spherical harmonics. SHTs were computed over (θ, ϕ) so that the input signals should be 1. INTRODUCTION first interpolated to spherical samples and spherical harmonics can only be evaluated on a fixed radius. The 3D rotation angle estimation problem deals with two 3D ob- Spherical Fourier transforms (SFTs) for rotational invariant fea- jects that own identical shapes but are aligned to different directions. tures were proposed by Wang et al. [9]. SFTs were defined by Given the samples of both objects, the 3D rotation angles between a 3D transform kernel with spherical harmonics in its θ- and ϕ- these objects are estimated. This issue plays a prominent role in pat- components. It was proved that SFTs are related to the Wigner-D tern recognition [1], computer vision [2], robotics [3], and comput- matrices for rotated objects and the performance would be better erized tomography imaging [4] when the orientation is of interest. than that of the spherical harmonics on a provided radius. However, Spherical harmonic transforms (SHTs) offer an approach to the this work only addressed the rotational invariance rather than rota- presented issue [1]. It is known that 3D rotation operations are re- tion angles between objects. In addition, SFTs are also transforms lated to the Wigner-D matrices in the spherical harmonic domain. on the spherical coordinates so Cartesian input samples are first con- The angle estimation problems are converted to estimating the pa- verted to spherical samples before applying SFTs. rameters of the Wigner-D matrices when the SHTs are given. Presented with the above issues, the angle estimation problem Discrete spherical harmonic oscillator transforms (discrete is reconsidered in this paper. For Cartesian samples, we utilized SHOTs) provide an efficient method to analyzing the spherical discrete SHOTs to perform this task. Discrete SHOTs analyze the components from the Cartesian samples without interpolation [5]. spherical components of volume data without any coordinate conver- Discrete SHOTs find useful applications in signal expansion, ro- sion. Besides, it will be proved that the angle estimation algorithms tational invariance analysis, and MRI data compression. Spherical proposed in [8] are compatible with discrete SHOTs, inspired by [9]. harmonics in the transform kernels make discrete SHOTs compatible Hence, higher accuracy could be expected when discrete SHOTs and with the algorithms stemming from the SHT. Based on such merit, the angle estimation formulae [8] are cascaded. this paper aims to combine discrete SHOTs with the angle estima- tion algorithm and to compare the performances of the estimators 3. PRELIMINARY based on spherical-harmonic-related transforms. This paper is organized as follows. Section 2 surveys the related 3.1. The rotation operation in three-dimensions work and Section 3 briefly reviews the concept of 3D rotations and Wigner-D matrices. Discrete SHOTs are introduced in Section4. It A 3D position vector r can be specified on the Cartesian coordinate is proved in Section 5 that discrete SHOTs are compatible with the system (x,y,z) R3 or on the spherical coordinate system, char- ∈ y y y 4. DISCRETE SPHERICAL HARMONIC OSCILLATOR TRANSFORMS ◦ 90◦ 90 x x x Spherical harmonic oscillator wavefunctions (SHOWs) originated from the wavefunctions of quantum harmonic oscillator systems in ◦ z z 90 z spherical coordinates [11]. Based on the system model, Schr¨odinger’s equation for this system is written as (a) X-rotation (b) Y-rotation (c) Z-rotation 1 2 2 3 Fig. 1. The rotation operation around (a) x-axis, (b) y-axis and (c) r r nℓm)= N + r nℓm) , (4) ◦ 2 −∇ h | 2 h | z-axis by 90 , respectively. The solid circles indicate the original objects while the empty circles show the new objects after rotation. where 2 denotes the 3D Laplacian operator and N = 2n + ℓ is the order of∇ the SHOWs. The bracket notations of SHOWs are denoted by r nℓm), indexing by integer parameters n, ℓ, and m, where h | acterized by the distance to the origin r, the polar angle θ, and the n, ℓ = 0, 1, 2,... ; m = ℓ, ℓ + 1,...,ℓ 1,ℓ. The physical − − − azimuthal angle ϕ. These parameters are defined over r [0, ), meaning of the bracket notation can be referred to [5, 10, 11] for the θ [0,π], and ϕ [0, 2π). The coordinates are uniquely∈ related∞ by interested readers. (4) is solved to be [11] ∈ ∈ x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ. ℓ ℓ+1/2 2 −r2/2 r nℓm)= Nnℓr L r e Yℓm(θ, ϕ), (5) The 3D rotation is specified by Euler angles [10]. Any 3D ro- h | n tation can be decomposed into the three basic rotation operations α where Nnℓ is the normalization factor related to n, ℓ and Ln ( ) are with respect to certain axes, as illustrated in Fig. 1. The Euler an- the associated Laguerre polynomials. SHOWs were proved to form· a gles cascade the basic rotations and parametrize the corresponding complete and orthonormal basis for L2 R3 , the set of finite-energy rotation angles. In this paper, the ZYZ convention for Euler angles signals in 3D [11]. is adapted. That is, the object is first rotated by α around z-axis, Alternatively, the wavefunctions for the quantum harmonic os- then followed by a rotation β around the y-axis in the rotated coor- cillator can also be solved on the Cartesian coordinates, i.e. solving dinate, and finally rotated by γ around the resultant z-axis, where (4) for r =(x,y,z), yielding separable Hermite Gaussian functions α, γ [0, 2π] and β [0,π]. The three Euler angles are described ∈ ∈ as the wavefunctions, by the operator α,β,γ , where the three rotation angles are specified 2 2 2 R − x +y +z in order. r 2 nxnynz = Ke Hnx (x)Hny (y)Hnz (z), (6) The Euler angles uniquely determine the 3D rotation except β = h | i 0,π. If β = 0, the rotation is equivalent to successive Z-rotations by where K denotes the normalization factor, Hn( ) are the Hermite · α and γ so that α + γ is unique rather than α and γ. On the other polynomials and the indices nx, ny, nz = 0, 1, 2,... The order of hand, β = π results in unique α γ. the separable Hermite Gaussian function is N = nx + ny + nz. Itis − obvious that (6) is separable in the Cartesian coordinates. Separable Hermite Gaussian functions are also complete and orthonormal basis 3.2. Spherical harmonic transforms and Wigner-D matrices for L2 R3 . In [12, 13], SHOWs and separable Hermite Gaussian func- Spherical harmonics are defined over (θ, ϕ) as tions are related to each other by the transformation coefficients n,ℓ,m Cnx,ny ,nz m 2ℓ + 1 (ℓ m)! m jmϕ Yℓm(θ, ϕ)=( 1) − Pℓ (cos θ) e , (1) r n,ℓ,m r − 4π (ℓ + m)! nℓm)= Cn ,n ,n nxnynz , (7) s h | x y z h | i where the summation runsX over all n , n , n with n + n + n = where ℓ = 0, 1, 2,..., ℓ m ℓ and P m( ) denote the associ- x y z x y z ℓ 2n + ℓ = N. (7) implies the SHOWs of order N are the finite linear ated Legendre polynomials.− ≤ Considering≤ a surface· signal f(θ, ϕ), its combination of the separable Hermite Gaussian functions of order spherical harmonic transforms (SHTs) F are the inner products ℓ,m N. Once Cn,ℓ,m is computed, the SHOWs, which are spherically between f(θ, ϕ) and Y (θ, ϕ) over the angles [8], nx,ny ,nz ℓm symmetric, can be synthesized from the separable Hermite Gaussian 2π π functions, which are separable in the Cartesian coordinates.